Throw a ball on a polygonal billiard table and it will trace a path
dictated by the laws of reflection. If the path loops back on itself
after n bounces, we call it a period-n billiard path. For a given
polygon, we are interested in counting all period-n billiard paths (up
to a natural equivalence relation). We will focus on the cases of the
rectangle, square, right isosceles triangle, and 30-60-90 triangle.
In the course of doing this, we will run across a geometric
interpretation of Euler's totient function and develop an intriguing
connection with integer partitions which hints at a much richer
structure than one would suspect.